Note:a1w denotes that first order adjoints are computed in working precision; this has the corresponding argument type nagad_a1w_w_rtype.
Also available is the t1w (first order tangent linear) mode, the interface of which is implied by replacing a1w by t1w throughout this document.
Additionally, the p0w (passive interface, as alternative to the FL interface) mode is available and can be inferred by replacing of active types by the corresponding passive types.
The method of codifying AD implementations in the routine name and corresponding argument types is described in the NAG AD Library Introduction.
is the adjoint version of the primal routine
Depending on the value of ad_handle,
c05ay_a1w_f uses algorithmic differentiation or symbolic adjoints to compute adjoints of the primal.
The routine may be called by the names c05ay_a1w_f or nagf_roots_contfn_brent_a1w. The corresponding t1w and p0w variants of this routine are also available.
is the adjoint version of the primal routine
c05ayf locates a simple zero of a continuous function in a given interval using Brent's method, which is a combination of nonlinear interpolation, linear extrapolation and bisection.
For further information see Section 3 in the documentation for c05ayf.
c05ay_a1w_f can provide symbolic adjoints by setting the symbolic mode as described in Section 3.2.2 in the X10 Chapter introduction. Please see Section 4 in the Introduction to the NAG AD Library for API description on how to use symbolic adjoints.
The symbolic adjoints assumes
(i)successful computation of primal problem ( or on exit of c05ayf), i.e.,
where is a solution;
(ii)the first derivative at the solution is not equal zero
In comparison to the algorithmic adjoint, the user-supplied primal and adjoint callbacks need specific implementation to support symbolic adjoint computation. Please see Section 4.5 in the Introduction to the NAG AD Library and recall what primal and adjoint callbacks need to calculate in case of an algorithmic adjoint.
Assuming the original user-supplied function evaluates
where is given by ruser or by use of COMMON globals. The variables and correspond to x of user-supplied subroutine f and return the value of f respectively.
The symbolic adjoint of c05ayf then requires in addition the following capability / modes:
(a)Function value evaluation only
(b)Function value evaluation and adjoint computation w.r.t. x only (corresponds to in the following equation), i.e.,
(c)Function value evaluation and adjoint computation w.r.t. only, i.e.,
Here is a placeholder for any user variable either passed via ruser or via COMMON global variables.
The library routine communicates via ad_config to the primal callback which version to provide, given the following API.
The symbolic adjoint computes
followed by an adjoint projection through the user-supplied adjoint routine
Both as well as are computed using the user-supplied adjoint routine.
You can set or access the adjoints of output argument x. The adjoints of all other output arguments are ignored.
c05ay_a1w_f increments the adjoints of the variable , where is given by the argument ruser or by use of COMMON globals (see (3)).
The adjoints of all other input parameters are not referenced.
Brent R P (1973) Algorithms for Minimization Without Derivatives Prentice–Hall
Du Toit J, Naumann U (2017) Adjoint Algorithmic Differentiation Tool Support for Typical Numerical Patterns in Computational Finance
Giles M (2017) Collected Matrix Derivative Results for Forward and Reverse Mode Algorithmic Differentiation
Naumann U, Lotz J, Leppkes K and Towara M (2017) Algorithmic Differentiation of Numerical Methods: Tangent and Adjoint Solvers for Parameterized Systems of Nonlinear Equations
In addition to the arguments present in the interface of the primal routine,
c05ay_a1w_f includes some arguments specific to AD.
A brief summary of the AD specific arguments is given below. For the remainder, links are provided to the corresponding argument from the primal routine.
A tooltip popup for all arguments can be found by hovering over the argument name in Section 2 and in this section.
c05ay_a1w_f preserves all error codes from c05ayf and in addition can return:
An unexpected AD error has been triggered by this routine. Please
See Section 4.8.2 in the NAG AD Library Introduction for further information.
Dynamic memory allocation failed for AD.
See Section 4.8.1 in the NAG AD Library Introduction for further information.
In symbolic mode the following may be returned:
8Parallelism and Performance
is not threaded in any implementation.
Please note that the algorithmic adjoint of Brent's method may be ill-conditioned. This means that derivatives of the zero returned in x, with respect to function parameters stored in ruser, may have limited accuracy when computed in algorithmic mode. This routine can be used in symbolic mode which will compute accurate derivatives.
The following examples are variants of the example for
modified to demonstrate calling the NAG AD Library.